Why Significant Digits Matter in Multiplication & Division

In summary, the conversation discusses the concept of significant digits and how they work in multiplication and division. It is explained that the result of a calculation is only as accurate as its least accurate input, and that even small errors can greatly affect the overall result. It is also mentioned that significant digits are an approximation, taking into account the maximum error in each factor.
  • #1
Diane_
Homework Helper
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A thread here asking about significant digits got me to thinking. I understand why they work the way they do with addition and subtraction, but multiplication and division are so easy that I've never bothered to think about it.

So - can someone explain why a three s.d. number times a two s.d. number gives a two s.d. number?
 
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  • #2
Garbage In-Garbage out! A chain is no stronger than its weakest link. The result of a calculation is no more accurate than its least accurate input!

Enough with the platitudes!

Suppose your three s.d number is 1.43 x 103 (in other words 1430 but writte so it is clear that we intend only 3 significant figures) and your 2 s.d. number is 4.8 x 10-2 (i.e. 0.048). The first number has been measured to the nearest "ten" and the second to the nearest "thousandth". What that means is that the first number could be as high as 1.435 x 103 or as low as 1.425 x 103. Similarly, the second number could be as high as 4.85 x 10-2 or as low as 4.75 x 10-2.

Suppose we actually made the full error, high, in both cases. Then the actual value of the product is 1.435 x 103*4.85 x 10-2= 6.95975 x 101= 69.5975. Suppose we made the full error, low, in both cases. Then the actual value of the product is 1.425 x 103*4.75 x 10-2= 6.76875 x 10 1= 67.6875.

That's a difference of 1.91 or about 2. Clearly saying that we know the value correct to the nearest 0.1 would be untrue. It's actually pushing it to say we have 2 s.f. since we aren't really certain that it must be between 67.5 and 68.5 which what "68" or "6.8 x 101" would mean but it is certainly much closer.
 
  • #3
So what you're saying is that it's basically an approximation, taking into account the maximum error in each factor?
 

1. Why do we need to use significant digits in multiplication and division?

Significant digits are used to indicate the precision of a measured or calculated value. In multiplication and division, the result should have the same number of significant digits as the number with the least number of significant digits in the calculation. This helps to avoid overestimating or underestimating the precision of the final result.

2. How do I determine the number of significant digits in a multiplication or division calculation?

The number of significant digits in a calculation is determined by the number with the least number of significant digits. For example, if you are multiplying 5.34 by 2.1, the number with the least number of significant digits is 2.1, which has two significant digits. Therefore, the final result should also have two significant digits.

3. What happens if I ignore significant digits in multiplication and division?

If you ignore significant digits in multiplication and division, you may end up with a result that is not precise or accurate. This is because significant digits are used to indicate the level of precision in a value. Ignoring significant digits may lead to overestimating or underestimating the precision of the final result.

4. Can I round my final result to the nearest whole number after a multiplication or division calculation?

No, rounding the final result to the nearest whole number after a multiplication or division calculation may result in a loss of precision. Instead, you should round the final result to the same number of significant digits as the number with the least number of significant digits in the calculation.

5. How do significant digits matter in scientific measurements?

In scientific measurements, significant digits are used to indicate the precision of a measured value. The more significant digits a value has, the more precise the measurement is. When performing calculations with measured values, it is important to maintain the same level of precision by using significant digits in order to obtain an accurate result.

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